scholarly journals Strong edge geodetic problem in networks

2017 ◽  
Vol 15 (1) ◽  
pp. 1225-1235 ◽  
Author(s):  
Paul Manuel ◽  
Sandi Klavžar ◽  
Antony Xavier ◽  
Andrew Arokiaraj ◽  
Elizabeth Thomas
Keyword(s):  
Graph Theory ◽  
Exact Solutions ◽  
Upper Bounds ◽  
Binary Trees ◽  
Lower And Upper Bounds ◽  
Transport Networks ◽  
Covering Problems ◽  
Geodetic Number ◽  
Block Graphs ◽  
Np Complete

Abstract Geodesic covering problems form a widely researched topic in graph theory. One such problem is geodetic problem introduced by Harary et al. [Math. Comput. Modelling, 1993, 17, 89-95]. Here we introduce a variation of the geodetic problem and call it strong edge geodetic problem. We illustrate how this problem is evolved from social transport networks. It is shown that the strong edge geodetic problem is NP-complete. We derive lower and upper bounds for the strong edge geodetic number and demonstrate that these bounds are sharp. We produce exact solutions for trees, block graphs, silicate networks and glued binary trees without randomization.

scholarly journals A Computational Perspective on Network Coding

2010 ◽  
Vol 2010 ◽  
pp. 1-11
Author(s):  
Qin Guo ◽  
Mingxing Luo ◽  
Lixiang Li ◽  
Yixian Yang
Keyword(s):  
Graph Theory ◽  
Computational Complexity ◽  
Lower Bound ◽  
Network Coding ◽  
Upper Bound ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Multicast Networks ◽  
Computational Perspective ◽  
Source Rate

From the perspectives of graph theory and combinatorics theory we obtain some new upper bounds on the number of encoding nodes, which can characterize the coding complexity of the network coding, both in feasible acyclic and cyclic multicast networks. In contrast to previous work, during our analysis we first investigate the simple multicast network with source rateh=2, and thenh≥2. We find that for feasible acyclic multicast networks our upper bound is exactly the lower bound given by M. Langberg et al. in 2006. So the gap between their lower and upper bounds for feasible acyclic multicast networks does not exist. Based on the new upper bound, we improve the computational complexity given by M. Langberg et al. in 2009. Moreover, these results further support the feasibility of signatures for network coding.

scholarly journals SOME BOUNDS ON SUM CONNECTIVITY AND PRODUCT CONNECTIVITY ZAGREB-K-BANHATTI INDICES OF GRAPHS.

2020 ◽  
Vol 9 (9) ◽  
pp. 144-157
Keyword(s):  
Graph Theory ◽  
Upper Bounds ◽  
Chemical Characteristics ◽  
Lower And Upper Bounds ◽  
Chemical Graph ◽  
Chemical Graph Theory ◽  
Connectivity Indices

The connectivity indices are applied to measure the chemical characteristics of compound in Chemical Graph Theory. In this paper, we introduce the sum connectivity Zagreb-K-Banhatti index and product connectivity Zagreb-K-Banhatti index of a graph. We provide lower and upper bounds for the sum connectivity Zagreb-K-Banhatti index and product connectivity Zagreb-K-Banhatti index of a graph in terms of Zagreb and K-Banhatti indices.

A GENERAL POSITION PROBLEM IN GRAPH THEORY

2018 ◽  
Vol 98 (2) ◽  
pp. 177-187 ◽  
Author(s):  
PAUL MANUEL ◽  
SANDI KLAVŽAR
Keyword(s):  
Graph Theory ◽  
Lower Bounds ◽  
General Position ◽  
Upper Bounds ◽  
Np Complete ◽  
Position Problem

The paper introduces a graph theory variation of the general position problem: given a graph $G$, determine a largest set $S$ of vertices of $G$ such that no three vertices of $S$ lie on a common geodesic. Such a set is a max-gp-set of $G$ and its size is the gp-number $\text{gp}(G)$ of $G$. Upper bounds on $\text{gp}(G)$ in terms of different isometric covers are given and used to determine the gp-number of several classes of graphs. Connections between general position sets and packings are investigated and used to give lower bounds on the gp-number. It is also proved that the general position problem is NP-complete.

scholarly journals NETWORK DECONTAMINATION IN PRESENCE OF LOCAL IMMUNITY

2007 ◽  
Vol 18 (03) ◽  
pp. 457-474 ◽  
Author(s):  
FABRIZIO LUCCIO ◽  
LINDA PAGLI ◽  
NICOLA SANTORO
Keyword(s):  
Upper Bound ◽  
Antiviral Agents ◽  
Antiviral Agent ◽  
Upper Bounds ◽  
Binary Trees ◽  
Cubic Graphs ◽  
Lower And Upper Bounds ◽  
Local Immunity ◽  
The Impact ◽  
Entire Network

We consider the problem of decontaminating a network infected by a mobile virus. The goal is to perform the task using as small a team of antiviral agents as possible, avoiding recontamination of disinfected areas. In all the existing literature, it is assumed that the immunity level of a decontaminated site is nil; that is, a decontaminated node, in absence of an antiviral agent on site, may be re-contaminated by any infected neighbour. The network decontamination problem is studied here under a new model of immunity to recontamination: we consider the case when a decontaminated vertex, after the cleaning agent has gone, will become recontaminated only if a majority of its neighbours are infected. We study the impact that the presence of local immunity has on the number of antiviral agents necessary to decontaminate the entire network. We establish both lower and upper bounds on the number cleaners in the case of (multidimensional) toroidal meshes, graphs of vertex degree at most three (e.g., cubic graphs, binary trees, etc.), and of tree networks. In all cases the established bounds are tight. All upper-bound proofs are constructive; i.e., we exhibit decontamination protocol achieving the claimed bound. We also analyze the total number of moves performed by the agents, and establish tight bounds in some cases.

scholarly journals On Sombor Index

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 140
Author(s):  
Kinkar Chandra Das ◽  
Ahmet Sinan Çevik ◽  
Ismail Naci Cangul ◽  
Yilun Shang
Keyword(s):  
Graph Theory ◽  
Topological Index ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Chemical Graph ◽  
Zagreb Indices ◽  
Chemical Graph Theory ◽  
Degree Of Vertex ◽  
Graph Parameters ◽  
Future Work

The concept of Sombor index (SO) was recently introduced by Gutman in the chemical graph theory. It is a vertex-degree-based topological index and is denoted by Sombor index SO: SO=SO(G)=∑vivj∈E(G)dG(vi)2+dG(vj)2, where dG(vi) is the degree of vertex vi in G. Here, we present novel lower and upper bounds on the Sombor index of graphs by using some graph parameters. Moreover, we obtain several relations on Sombor index with the first and second Zagreb indices of graphs. Finally, we give some conclusions and propose future work.

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

Energy of spherical fuzzy graphs

2020 ◽  
pp. 321-332
Author(s):  
S. Yahya Mohamed ◽  
A. Mohamed Ali
Keyword(s):  
Adjacency Matrix ◽  
Upper Bounds ◽  
Fuzzy Graph ◽  
Lower And Upper Bounds ◽  
Fuzzy Graphs

In this paper, the notion of energy extended to spherical fuzzy graph. The adjacency matrix of a spherical fuzzy graph is defined and we compute the energy of a spherical fuzzy graph as the sum of absolute values of eigenvalues of the adjacency matrix of the spherical fuzzy graph. Also, the lower and upper bounds for the energy of spherical fuzzy graphs are obtained.

scholarly journals Relationship between Age and Value of Information for a Noisy Ornstein–Uhlenbeck Process

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 940
Author(s):  
Zijing Wang ◽  
Mihai-Alin Badiu ◽  
Justin P. Coon
Keyword(s):  
Value Of Information ◽  
Markov Models ◽  
Upper Bounds ◽  
Distribution Functions ◽  
Lower And Upper Bounds ◽  
Uhlenbeck Process ◽  
Source Data ◽  
Non Linear ◽  
Queue Model ◽  
Selection Of

The age of information (AoI) has been widely used to quantify the information freshness in real-time status update systems. As the AoI is independent of the inherent property of the source data and the context, we introduce a mutual information-based value of information (VoI) framework for hidden Markov models. In this paper, we investigate the VoI and its relationship to the AoI for a noisy Ornstein–Uhlenbeck (OU) process. We explore the effects of correlation and noise on their relationship, and find logarithmic, exponential and linear dependencies between the two in three different regimes. This gives the formal justification for the selection of non-linear AoI functions previously reported in other works. Moreover, we study the statistical properties of the VoI in the example of a queue model, deriving its distribution functions and moments. The lower and upper bounds of the average VoI are also analysed, which can be used for the design and optimisation of freshness-aware networks. Numerical results are presented and further show that, compared with the traditional linear age and some basic non-linear age functions, the proposed VoI framework is more general and suitable for various contexts.

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